The Reds extended their unbeaten streak to a seventh match in some style at Craven Cottage, twice coming from behind before eking out a 3-2 victory courtesy of Steven Gerrard's stoppage-time penalty.
The victory moves them within a point of third-placed Manchester City, whose clash with Sunderland was postponed due to the bad weather, and four points off leaders Chelsea.
Jose Mourinho recently described his side as a "little horse" that were not ready to win the title - an argument dismissed by City boss Manuel Pellegrini and laughed off by Rodgers.
The Liverpool boss joked his side "may be the chihuahua's that run in between the horses' legs" and preferred to focus attention on his side's impressive mental strength.
"It was a very important win for us tonight," Rodgers said. "Obviously after the game at the weekend when we were outstanding [beating Arsenal 5-1], we had to show some different qualities tonight.
"One of them was character and this was a wonderful demonstration of the wonderful togetherness of the group.
"You're always pushing and you're always looking to get something. For me, it's never over until the whistle goes.
"Yes, they were disappointing goals. We've defended poorly at times this season, it has not been structurally but mistakes and individual errors.
"But thankfully we're a very aggressive and offensive team going forward, so we have goals in the team.
"That's something I've been trying to build since I've been here. When I arrived into the club, the complaint about the team was that they didn't score enough.
"Now we know we can go to any ground, or at home, and get goals. Obviously tonight the goals got us the three points rather than our defending."
It had looked like Liverpool would be undone by defensive mistakes, with Kolo Toure's calamitous own goal handing Fulham an eighth-minute lead.
Daniel Sturridge levelled after meeting an exquisite first-time Gerrard through ball, only for another defensive mistake, this time by Martin Skrtel, to allow Kieran Richardson to restore the hosts' lead.
The Reds drew parity once more through Philippe Coutinho before Gerrard struck home a stoppage-time penalty to secure victory and Rodgers' praise.
"Steven is peerless in many aspects of football," the Liverpool boss said of his captain.
"He was brilliant against Arsenal, dictating the game, playmaking from behind and his pass for the first goal here.
"He has that penetration he has shown all his life and then his composure at the end was not by accident. He studies the goalkeeper, which way he dives, so that professionalism is still there.
"There's no greater person than Steven Gerrard to take a penalty right at the end to get the three points and he tucked it away really well."
Fulham boss Rene Meulensteen revealed Sascha Riether apologised to his team-mates for the challenge on Sturridge, from which Gerrard netted the winner.
The Dutchman was understandably frustrated by the loss but, given Sunday's 2-2 draw at Manchester United, was proud of his side's display against the in-form Reds.
"I definitely think we deserved a point," Meulensteen said. "If you look at the game, we got our noses in front again.
"We actually played quite well, were well organised, we didn't let them settle into a rhythm. They didn't cause us too much danger.
"Out of a nothing ball, Gerrard made a pass with the quality he's got and they stuck it in the net so you have to start again.
"We did, we took the chance which fell for us to see us go 2-1, but they came back.
"But what you will see from a team in this position is that they will try to hold on more than build on it, so you drop a little bit deeper, get the balls in the right area and stop them hurting you.
"That was the case with the Coutinho goal, letting him come inside and then there was a deflection. Then the third one was just a silly challenge that should not have been made by Sascha.
"The first thing he admitted when he walked in the dressing room was: 'Sorry, I shouldn't have done it'.
"But when you lose 3-2, which is very hard to take with the effort the boys put in they should have definitely earned a point."